[latexpage]
Simplifying rational exponents equations that are more difficult generally involves two steps. First, reduce inside the brackets. Second, multiplu the power outside the brackets for all terms inside.
Example 1
Simplify the following rational exponent expression:
\(\left(\dfrac{x^{-2}y^{-3}}{x^{-2}y^4}\right)^2\)
First, simplifying inside the brackets gives:
\(x^{-2–2}y^{-3-4}\)
Or:
\(x^0y^{-7}\)
Which simplifies to:
\(y^{-7}\)
Second, taking the exponent 2 outside the brackets and applying it to the reduced expression gives:
\(y^{-7\cdot 2} \text{ or }y^{-14}\)
Therefore:
\(\left(\dfrac{x^{-2}y^{-3}}{x^{-2}y^4}\right)^2=y^{-14}\)
Example 2
Simplify the following rational exponent expression:
\(\left(\dfrac{x^{-4}y^{-6}}{x^{-5}y^{10}}\right)^{-3}\)
First, simplifying inside the brackets gives:
\(x^{-4–5}y^{-6-10}\)
Or:
\(x^1y^{-16}\)
Which simplifies to:
\(xy^{-16}\)
Second, taking the exponent −3 outside the brackets and applying it to the reduced expression gives:
\((xy^{-16})^{-3}\text{ or }x^{-3}y^{48}\)
Therefore:
\(\left(\dfrac{x^{-4}y^{-6}}{x^{-5}y^{10}}\right)^{-3}=x^{-3}y^{48}=\dfrac{y^{48}}{x^3}\)
Example 3
Simplify the following rational exponent expression:
\(\left(\dfrac{a^0b^3}{c^6d^{-12}}\right)^{\frac{1}{3}}\)
First, simplifying inside the brackets gives:
\(\dfrac{b^3}{c^6d^{-12}}\)
Second, taking the exponent \(\frac{1}{3}\) outside the brackets and applying it to the reduced expression gives:
\(\dfrac{b^{3\cdot \frac{1}{3}}}{c^{6\cdot \frac{1}{3}}d^{-12\cdot \frac{1}{3}}}\)
Or:
\(\dfrac{b}{c^2d^{-4}}\)
Which simplifies to:
\(\dfrac{bd^4}{c^2}\)
Questions
Simplify the following rational exponents.
- \(\left(\dfrac{x^{-2}y^{-6}}{x^{-2}y^4}\right)^2\)
- \(\left(\dfrac{x^{-3}y^{-3}}{x^{-1}y^6}\right)^3\)
- \(\left(\dfrac{x^{-2}y^{-4}}{x^2y^{-4}}\right)^2\)
- \(\left(\dfrac{x^{-5}y^{-3}}{x^{-4}y^2}\right)^4\)
- \(\left(\dfrac{x^{-2}y^{-2}}{x^{-3}y^3}\right)^8\)
- \(\left(\dfrac{x^{-4}y^{-3}}{x^{-3}y^2}\right)^5\)
- \(\left(\dfrac{x^{-2}y^{-4}}{x^{-2}y^4}\right)^{-2}\)
- \(\left(\dfrac{x^{-2}y^{-3}}{x^{-5}y^3}\right)^{-3}\)
- \(\left(\dfrac{x^{-2}y^{-3}}{x^{-2}y^{-3}}\right)^{-1}\)
- \(\left(\dfrac{x^{-2}y^{-3}}{x^{-2}y^4}\right)^{-2}\)
- \(\left(\dfrac{x^0y^{-3}}{x^{-2}y^0}\right)^{-5}\)
- \(\left(\dfrac{x^{-22}y^{-36}}{x^{-24}y^{12}}\right)^0\)
- \(\left(\dfrac{a^0b^3}{a^6b^{-12}}\right)^{-\frac{1}{3}}\)
- \(\left(\dfrac{a^{12}b^4}{a^8c^{-12}}\right)^{\frac{1}{4}}\)
- \(\left(\dfrac{a^5c^{10}}{b^5d^{-15}}\right)^{\frac{2}{5}}\)
- \(\left(\dfrac{a^2b^8}{a^6b^{-12}}\right)^{-\frac{3}{4}}\)
- \(\left(\dfrac{a^0b^3}{c^6d^{-12}}\right)^{\frac{0}{3}}\)
- \(\left(\dfrac{a^0b^3}{c^6d^{-12}}\right)^{\frac{1}{10}}\)
Answers to odd questions
1. \((x^{-2–2}y^{-6-4})^2\)
\((1\cancel{x^0}y^{-10})^2\)
\(y^{-20}\text{ or }\dfrac{1}{y^{20}}\)
3. \((x^{-2-2}y^{-4–4})^2\)
\((x^{-4}\cancel{y^0}1)^2\)
\(x^{-8}\text{ or }\dfrac{1}{x^8}\)
5. \((x^{-2–3}y^{-2-3})^8\)
\((x^1y^{-5})^8\)
\(x^8y^{-40}\text{ or }\dfrac{x^8}{y^{40}}\)
7. \((x^{-2–2}y^{-4-4})^{-2}\)
\((1\cancel{x^0}y^{-8})^{-2}\)
\(y^{16}\)
9. \((x^{-2–2}y^{-3–3})^{-1}\)
\((\cancel{x^0y^0}1)^{-1}\)
1
11. \(\left(\dfrac{1\cancel{x^0}y^{-3}}{x^{-2}\cancel{y^0}1}\right)^{-5} \\ \)
\(\left(\dfrac{x^2}{y^3}\right)^{-5} \\ \)
\(\dfrac{x^{-10}}{y^{-15}} \\ \)
\(\dfrac{y^{15}}{x^{10}}\)
13. \(\left(\dfrac{1 \cancel{a^0}b^3}{a^6b^{-12}}\right)^{-\frac{1}{3}} \\ \)
\(\left(\dfrac{b^15}{a^6}\right)^{-\frac{1}{3}} \\ \)
\(\dfrac{b^-5}{a^-2}\text{ or }\dfrac{a^2}{b^5}\)
15. \(\left(\dfrac{a^5c^{10}d^{15}}{b^5}\right)^{\frac{2}{5}} \\ \)
\(\dfrac{a^2c^4d^6}{b^2}\)
17. 1
